1 Reverend Bayes updates our Believe in Flood Detection

How an 275 year old idea helps map the extent of floods

Code

%matplotlib widget

import numpy as np
from eomaps import Maps

from tuw_education_notebooks.view_bayes_flood import view_bayes_flood
from tuw_education_notebooks.calc_bayes_flood import sig0_dc, calc_likelihoods

1.1 From Backscattering to Flood Mapping

This notebook explains how microwave (\(\sigma^0\)) backscattering (Figure 1.1) can be used to map the extent of a flood. We replicate in this exercise the work of Bauer-Marschallinger et al. (2022) on the TU Wien Bayesian-based flood mapping algorithm.

In the following lines we create a map with EOmaps (Quast 2024) of the \(\sigma^0\) backscattering values.

Code

m = Maps(ax=121, crs=3857)
m.set_data(data=sig0_dc, x="x", y="y", parameter="SIG0", crs=Maps.CRS.Equi7_EU)
m.plot_map()
m.add_colorbar(label="$\sigma^0$ (dB)", orientation="vertical")
m.add_scalebar(n=5)
m2 = m.new_map(ax=122, crs=3857)
m2.set_extent(m.get_extent())
m2.add_wms.OpenStreetMap.add_layer.default()
m.apply_layout(
    {
        'figsize': [7.5, 5.25],
        '0_map': [0.05, 0.18, 0.35, 0.64],
        '1_cb': [0.75, 0.1, 0.1, 0.8],
        '1_cb_histogram_size': 0.8,
        '2_map': [0.4, 0.18, 0.35, 0.64]
        }
    )
m.show()

Figure 1.1: Area targeted for \(\sigma^0\) backscattering is the Greek region of Thessaly, which experienced a major flood in February of 2018.

1.2 Microwave Backscattering over Land and Water

Reverend Bayes was concerned with two events, one “the hypothesis” occurring before the other “the evidence”. If we know the cause it is easy to logically deduce the probability of an effect. But in this case we want to deduce the probability of a cause from an observed effect, also known as “reversed probability”. In the case of flood mapping we have \(\sigma^0\) backscatter observations over land (effect) and we want to deduce the probability of flooding (F) and non-flooding (NF).

Phrased differently we want to know the probability of flooding given a pixel’s \(\sigma^0\):

\[P(F|\sigma^0)\]

and the probability of a pixel being not flooded given a certain \(\sigma^0\):

\[P(NF|\sigma^0).\]

Bayes showed that these can be deduced from the observation that forward and reversed probability are equal, so that:

\[P(F|\sigma^0)P(\sigma^0) = P(\sigma^0|F)P(F)\]

and

\[P(NF|\sigma^0)P(\sigma^0) = P(\sigma^0|NF)P(NF).\]

The forward probability of \(\sigma^0\) given flood (\(P(\sigma^0|F)\)) and \(\sigma^0\) given non-flood (\(P(\sigma^0|NF)\)) can be extracted from past information on backscattering over land and water surfaces. As seen in the sketch below (Figure 1.2), the characteristics of backscattering over land and water differ considerably.

Figure 1.2: Schematic backscattering over land and water. Image from Geological Survey Ireland

1.3 Likelihoods

These so-called likelihoods of \(P(\sigma^0|F)\) and \(P(\sigma^0|NF)\) can thus be calculated from past backscattering information. Without going into the details of how these likelihoods are calculated, you can click on a pixel of the map to plot the likelihoods of \(\sigma^0\) being governed by land or water.

Code

view_bayes_flood(sig0_dc)

Figure 1.3: Likelihoods for \(\sigma^0\) being associated with land or water for 1 pixel in the Greek area of Thessaly. Likelihoods are calculated over a range of \(\sigma^0\). The pixel’s observed \(\sigma^0\) is given with a vertical line. Click on the map to re-calculate and update this figure for another pixel in the study area. Map created with EOmaps (Quast 2024).

1.4 Posteriors

Having calculated the likelihoods, we can now move on to calculate the probability of (non-)flooding given a pixel’s \(\sigma^0\). These so-called “posteriors” need on more piece of information as can be seen in the equation above. We need the probability whether a pixel is flooded \(P(F)\) or non-flooded \(P(NF)\). We can of course not estimate this information precisely, so our best estimate is that the chance of flooding versus non-flooding of a pixel is 50-50: a coin flip. This subjective prior believe is a fundament of Bayesian statistics and we use the likelihood to update our belief in this particular hypothesis. We now can also calculate the probability of backscattering \(P(\sigma^0)\), as the weighted average of the water and land likelihoods. The latter ensures that our updated evidence (or posteriors) ranges between 0 to 1.

The following code block shows how we calculate the priors.

Code

def calc_posteriors(water_likelihood, land_likelihood):
    evidence = (water_likelihood * 0.5) + (land_likelihood * 0.5)
    return (water_likelihood * 0.5) / evidence,  (land_likelihood * 0.5) / evidence

We can plot these posteriors of the probability of flooding and non-flooding again and compare these to pixel’s measured \(\sigma^0\). Click on a pixel to calculate the posterior probability.

Code

view_bayes_flood(sig0_dc, calc_posteriors)

Figure 1.4: Posterior probabilities for \(\sigma^0\) of 1 pixel being associated with land for water in the Greek area of Thessaly. Click on the map to re-calculate and update this figure for another pixel in the study area. Map created with EOmaps (Quast 2024).

1.5 Flood Classification

We are now ready to combine all this information and classify the pixels according to the probability of flooding given the backscatter value of 1 pixel. Here we just look whether the probability of flooding is higher than non-flooding:

Code

def bayesian_flood_decision(id, sig0_dc):
    nf_post_prob, f_post_prob = calc_posteriors(*calc_likelihoods(id,  sig0_dc))
    return np.greater(f_post_prob, nf_post_prob)

Click on a point in the below map to see the likelihoods and posterior distributions (in the left-hand subplots).

Code

view_bayes_flood(sig0_dc, calc_posteriors, bayesian_flood_decision)

Figure 1.5: Flood extent of the Greek region of Thessaly based on Bayesian probabilities are shown on the map superimposed on an open street map. Click on a pixel to generate the point’s water and land likelihoods as well as the posterior probabilities. Map created with EOmaps (Quast 2024).